TL;DR:
Though the setting is technically scifi, it makes more sense to view it as fantasy. There are dragons -- they come in five colors, two of them shiny. There are crafts guilds called Halls. Most people farm. There are some Lords -- no kings, but the Lord Holders are basically kings. There’s a formless, flesh-eating menace that appears from the sky when a certain cosmic event recurs -- the arrival on the horizon of the baleful Red Star.
The planet Pern is a lot like Earth – its intended use? A sort of a hippie agrarian commune for human colonists wanting to escape interstellar politics.
These colonists landed and got their society started... only to be completely blindsided by the aforementioned menace from the sky. It’s called Thread, it falls for a couple decades every two centuries or so, and it totally destroys anything organic -- animal, plant, anything -- though it dissolves in water and can be burned. Not having the industrial capacity to manufacture enough flamethrowers, the Pernese decided to make big organic breedable flying flamethrowers (conveniently stuck to human beings, to keep them in check) to kill the Thread. The intelligent and majestic dragons were genetically engineered from little dragon-looking critters called firelizards.
There was a slow regression of technology from futuristic to essentially late-medieval European. A tripartite governmental system emerged: Hold, Hall, and Weyr working together. Essentially, Holds are cities/towns, Halls are craft guilds, and the Weyrs are where dragons and dragonriders live. Though the Holds try to disagree, the basis of the Pernese economy is the support of the Weyrs; dragons themselves eat a lot of meat, which has to be raised on valuable farmland, and the humans of the Weyr require a lot of material support in terms of craft goods and feeding. Hold, Hall, and Weyr support and check each other, preventing despotism or strongmaning.
Culturally, you can pretty much imagine the Pern aligns to its rough technological level -- in many ways, it is quite like medieval Europe. Because of the quick travel capabilities of dragons, culture is pretty homogeneous across the people of Pern (canonically to an incredibly unrealistic degree), and everyone speaks the same language and shares the same basic cultural ideals. As an interesting wrinkle, Pern has no religion. There is superstition -- it’s hard to be human without it -- but the role of The Church in medieval Europe is instead invested in the Weyrs. The people of Pern idolize dragonriders in a similar way to the idolization of clergy and saints in the medieval period.
Human rights are a bit shaky on Pern. There isn’t much racism (thanks to the initial colonists rejecting colorism and the fact that everyone shares the same basic culture), but sexism, queerphobia, and classism are very common. Women are not necessarily seen as the lesser sex, but have fewer opportunities in life: some of the Hallcrafts rarely accept women, and most dragonriders are men (depending on the Weyr). Queer identities are generally looked down upon, sometimes violently. Classism is rampant: Lord Holders live like kings, master craftspersons wield considerable influence and funds, while the common farmers live in huts and many struggle to support themselves.
A quick note on how Weyrs are different: because of the way dragon bonding works (it's essentially random selection), people from all walks of life end up as dragonriders and this makes the Weyrs bastions of tolerance and egalitarianism compared to other places. Also, the Weyrs are sexually liberal, also because dragons. They also tend to have a higher standard of living.